The purpose of this study is to develop thymic transplantation as a therapy for partial DiGeorge syndrome. The patient population includes only those patients with DiGeorge syndrome who have low numbers of T cells and poor T cell function after one year of age. The patients are not treated with thymic transplantation during the first year of life because there is the possibility of spontaneous improvement during this time. Human postnatal thymic tissue is obtained after informed consent from infants undergoing heart surgery. (This thymic tissue normally would be discarded.) The thymic tissue is cultured in the laboratory for 2 weeks prior to transplant to allow time for safety testing of the donor (for EBV, CMV, hepatitis B and C, and HIV 1 and 2) and to allow mature T cells in the donor thymus to die. The thymus is transplanted into the quadriceps muscle of the recipient. The patient is followed after transplantation by periodic assessment of T cell numbers and function. T cell function is assessed by response of peripheral blood mononuclear cells to mitogens, by mixed lymphocyte reactions, and by in vitro responses after immunization with tetanus toxoid. A biopsy of the transplanted thymus is done at 3 months after the transplant. Monoclonal antibody analysis of frozen sections of the biopsy are used to determine whether there is host thymopoiesis occurring in the donor epithelium. HLA antibodies distinguish donor from host cells.